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Explaining Volatility Smiles of Equity Options
with Capital Structure Models
Swiss Institute of Banking and Finance
University of St. Gallen
Institut for Regnskab og Finansiering
May 31, 2005
We investigate the behavior of prices of equity options in the Cor-porate Securities Framework suggested by Ammann and Genser (2004)where equity is the residual claim of a stochastic EBIT. Option prices areobtained by two numerical methods: (i) Approximation of the EBIT-process by a trinomial lattice and calculation of all securities by back-ward induction. (ii) Evaluation of the risk-neutrally expected value ofthe equity option at maturity by direct numerical integration. Econom-ically, the current state of the firm with respect to bankruptcy and thecapital structure influence to a large extent the particular risk-neutralequity (return) distribution at option maturity and the level and slopeof implicit Black and Scholes (1973)-volatilities as a function of strikeprices. Additionally, we oppose the tradition of relating equity returnmoments to implied volatilities. This connection might be misleading
Michael Genser (Michael.Genser@unisg.ch) is visiting researcher at the Department ofAccounting and Finance, University of Southern Denmark, Campusvej 55, DK-5320 OdenseM. I would like to thank seminar participants at the University of Southern Denmark inOdense, Paul Soderlind, Rico von Wyss, Stephan Kessler, Ralf Seiz, Michael Verhofen, BerndGenser, Bernd Brommundt, and especially Christian Riis flor for helpful comments andinsightful discussion. Financial support by the National Centre of Competence in ResearchFinancial Valuation and Risk Management (NCCR FINRISK) is gratefully acknowledged.NCCR FINRISK is a research program supported by the Swiss National Science Foundation.
when bankruptcy probabilities become high.JEL-Classification: G13, G33Keywords: option pricing; volatility smiles; firm value models; bankruptcy
Explaining Volatility Smiles of Equity Options withCapital Structure Models
We investigate the behavior of prices of equity options in the Cor-porate Securities Framework suggested by Ammann and Genser (2004)where equity is the residual claim of a stochastic EBIT. Option pricesare obtained by two numerical methods: (i) Approximation of the EBIT-process by a trinomial lattice and calculation of all securities by backwardinduction. (ii) Evaluation of the risk-neutrally expected value of the eq-uity option at maturity by direct numerical integration. Economically,the current state of the firm with respect to bankruptcy and the capitalstructure influence to a large extent the particular risk-neutral equity (re-turn) distribution at option maturity and the level and slope of implicitBlack and Scholes (1973)-volatilities as a function of strike prices. Addi-tionally, we oppose the tradition of relating equity return moments to im-plied volatilities. This connection might be misleading when bankruptcyprobabilities become high.JEL-Classification: G13, G33Keywords: option pricing; volatility smiles; firm value models; bankruptcy
One of the frequently discussed issues in the asset pricing literature iswhy the theoretical option prices in the Black and Scholes (1973)/Merton(1974) framework cannot be observed empirically. Implied volatilities ofobserved option prices calculated in the Black/Scholes setting are notconstant. They are higher for lower strike prices than for higher ones.The convex relationship is commonly referred to as an options volatilitysmile or smirk.1
1The literature on implied option volatilities is huge. Early evidence of the existence ofimplied volatility smiles is MacBeth and Merville (1979) who base their work on studies ofLatane and Rendleman (1975) and Schmalensee and Trippi (1978). Emanuel and MacBeth(1982) try to relax the stringent volatility assumption of the Black and Scholes (1973) toaccount for the volatility smile but fail by a constant elasticity of variance model of thestock price. More recent studies of Rubinstein (1994), Jackwerth and Rubinstein (1996),and Jackwerth and Rubinstein (2001) take the volatility smile as given and exploit optionprices to extract implied densities of the underlying asset. See also Buraschi and Jackwerth(2001) who report that after the 1987 market crash, the spanning properties of optionsdecreased. They conclude that more assets are needed for hedging option prices which hintsto additional risk factors such as stochastic volatility.
Several extensions of the Black/Scholes framework have been sug-gested to account for these empirical observations which can be catego-rized in two groups. First, a pragmatic stream of the literature intro-duced volatility structures and thus changed the physical distributionalassumptions for the underlying. Although this procedure yields satisfyingresults for equity option trading, economic intuition is still lacking whichunderpins the use of volatility structures.2
Second, an economic stream of the literature tried to explain why thepricing kernel, defined as the state price function at option maturity, isdifferent to the one implied by the Black and Scholes (1973) model. Arrow(1964) and Debreu (1959), Rubinstein (1976), Breeden and Litzenberger(1978), Brennan (1979) and others relate state prices to investor utilityand the state dependent payoffs of securities and disentangle the effectof the investors utility function from the probability distribution of theunderlying asset. Therefore, the pricing kernel and the value of securi-ties depends on assumptions about the utility function of the investorand on the distribution of the security.3 The pricing kernel is valid for allsecurities in an economy. However, when taking an individual firms per-spective, the simple pricing kernel needs to be augmented by additionalrisk factors such as default or liquidity.
Using the Corporate Securities Framework of Genser (2005a) and theanalytical solution of Ammann and Genser (2005) and Genser (2005b) asimpler economic explanation might be suggested: The implied Black/Scholes-volatility smile of equity options can be related to the specific ability ofequity holders to declare bankruptcy. This feature introduces dependenceon the particular path of EBIT and changes the distribution of equity dueto the conditioning on survival until option maturity. Contract design ofequity, especially the limited liability, changes the local volatility of eq-uity which in turn depends on the current state of the firm and influencesthe pricing kernel and equity options implied volatilities. Our approachis related to Geske (1979)s compound option approach. However, in con-trast to Geske (1979) whose underlying of the compound option mightbe interpreted as a Merton (1974)-like firm with only one finite maturityzero bond outstanding, the Corporate Securities Framework allows for
2See e.g. Rubinstein (1994) who adopts this procedure to the binomial model by allowingan arbitrary distribution of the underlying at option maturity. Heston and Nandi (2000)derive closed form solutions for options where the volatility of the underlying which followsa GARCH process.
3Franke, Stapleton and Subrahmanyam (1999) analyze the pricing kernel directly bycomparing pricing kernels of investors with changing degree of risk aversion at differentlevels of investors wealth. Franke et al. (1999) find the pricing kernel to be convex toaccommodate for the dependence of risk aversion and investors wealth leading to conveximplied Black/Scholes-volatilities.
a complex capital structure. As will be shown later, the debt structureinfluences the slope and the level of the implied volatility smile.4
In a related empirical study, Bakshi, Kapadia and Madan (2003) linkthe volatility smile to the distribution of equity returns. They show thata higher skewness and a lower kurtosis of equity returns result in steepervolatility smiles. Empirical evidence supports their hypothesis. However,Bakshi et al. (2003) do not offer an economic explanation why individualstocks returns should be skewed. In our EBIT-based firm value frame-work, the leverage ratio depends on the current state of the firm withrespect to bankruptcy. Firms far from bankruptcy and with low leverageratios exhibit a risk-neutral equity distribution that reflects the proper-ties of the assumed EBIT-process. The function of implied equity optionvolatilities with respect to strike prices are at a low level but steep. Thecloser the firm is to bankruptcy skewness and kurtosis of equity valuesincrease. The implied volatility level rises significantly but the smile be-comes flatter, at least in the ABM-case. However, we stress that highermoments of equity returns might be misinterpreted in the presence ofbankruptcy probabilities. Moreover, a key determinant of implied volatil-ity structures is the firms capital structure.
Toft and Prucyk (1997) value equity options in the restrictive Le-land (1994) framework. The Corporate Securities Framework extendsthe Toft and Prucyk (1997) analysis. Their model is a special case of ourframework if we restrict the capital structure to perpetual debt, the taxstructure to corporate taxes only, and if we assume that free cash flow af-ter taxes follows a geometric Brownian motion instead of EBIT followingan arithmetic or geometric Brownian motion. We are able to analyze thepricing of options under different assumptions for the EBIT-process andof firms that have a complex capital structure. Toft and Prucyk (1997)shed some light an complex capital structures when they proxy a debtcovenant in the perpetual debt case by the amount of a firms short termdebt. Our model allows to analyze firms with short term debt and